# American Institute of Mathematical Sciences

June  2012, 5(2): 283-323. doi: 10.3934/krm.2012.5.283

## Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension

 1 Istituto per le Applicazioni del Calcolo (sezione di Bari), via G. Amendola, 122/D, 70126 BARI, Italy

Received  April 2011 Revised  November 2011 Published  April 2012

In the kinetic theory of gases, a class of one-dimensional problems can be distinguished for which transverse momentum and heat transfer effects decouple. This feature is revealed by projecting the linearized Boltzmann model onto properly chosen directions (which were originally discovered by Cercignani in the sixties) in a Hilbert space. The shear flow effects follow a scalar integro-differential equation whereas the heat transfer is described by a $2 \times 2$ coupled system. This simplification allows to set up the well-balanced method, involving non-conservative products regularized by solutions of the stationary equations, in order to produce numerical schemes which do stabilize in large times and deliver accurate approximations at numerical steady-state. Boundary-value problems for the stationary equations are solved by the technique of "elementary solutions" at the continuous level and by means of the "analytical discrete ordinates" method at the numerical one. Practically, a comparison with a standard time-splitting method is displayed for a Couette flow by inspecting the shear stress which must be a constant at steady-state. Other test-cases are treated, like heat transfer between two unequally heated walls and also the propagation of a sound disturbance in a gas at rest. Other numerical experiments deal with the behavior of these kinetic models when the Knudsen number becomes small. In particular, a test-case involving a computational domain containing both rarefied and fluid regions characterized by mean free paths of different magnitudes is presented: stabilization onto a physically correct steady-state free from spurious oscillations is observed.
Citation: Laurent Gosse. Well-balanced schemes using elementary solutions for linear models of the Boltzmann equation in one space dimension. Kinetic & Related Models, 2012, 5 (2) : 283-323. doi: 10.3934/krm.2012.5.283
##### References:
 [1] D. Amadori, L. Gosse and G. Guerra, Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws,, Arch. Rational Mech. Anal., 162 (2002), 327. doi: 10.1007/s002050200198. Google Scholar [2] K. Aoki and C. Cercignani, A technique for time-dependent boundary value problems in the kinetic theory of gases. I. Basic analysis,, Z. Angew. Math. Phys., 35 (1984), 127. doi: 10.1007/BF00947927. Google Scholar [3] J. Appell, A. S. Kalitvin and P. P. Zabrejko, Boundary value problems for integro-differential equations of Barbashin type,, J. Integral Equ. Applic., 6 (1994), 1. doi: 10.1216/jiea/1181075787. Google Scholar [4] A. Arnold, J. A. Carrillo and M. D. Tidriri, Large-time behavior of discrete equations with non-symmetric interactions,, Math. Mod. Meth. in Appl. Sci., 12 (2002), 1555. Google Scholar [5] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations,, J. Stat. Phys., 63 (1991), 323. doi: 10.1007/BF01026608. Google Scholar [6] L. B. Barichello, M. Camargo, P. Rodrigues and C. E. Siewert, Unified solutions to classical flow problems based on the BGK model,, Z. Angew. Math. Phys., 52 (2001), 517. doi: 10.1007/PL00001559. Google Scholar [7] L. B. Barichello and C. E. Siewert, A discrete-ordinates solution for a non-grey model with complete frequency redistribution,, JQSRT, 62 (1999), 665. Google Scholar [8] G. R. Bart and R. L. Warnock, Linear integral equations of the third kind,, SIAM J. Math. Anal., 4 (1973), 609. doi: 10.1137/0504053. Google Scholar [9] P. Bassanini, C. Cercignani and C. D. Pagani, Comparison of kinetic theory analyses of linearized heat transfer between parallel plates,, Int. J. Heat Mass Transfer, 10 (1967), 447. doi: 10.1016/0017-9310(67)90165-2. Google Scholar [10] P. Bassanini, C. Cercignani and C. D. Pagani, Influence of the accommodation coefficient on the heat transfer in a rarefied gas,, Int. J. Heat Mass Transfer, 11 (1968), 1359. doi: 10.1016/0017-9310(68)90181-6. Google Scholar [11] R. Beals, An abstract treatment of some forward-backward problems of transport and scattering,, J. Funct. Anal., 34 (1979), 1. doi: 10.1016/0022-1236(79)90021-1. Google Scholar [12] R. Beals and V. Protopopescu, Half-range completeness for the Fokker-Planck equation,, J. Stat. Phys., 32 (1983), 565. doi: 10.1007/BF01008957. Google Scholar [13] M. Bennoune, M. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics,, J. Comp. Phys., 227 (2008), 3781. doi: 10.1016/j.jcp.2007.11.032. Google Scholar [14] A. Biryuk, W. Craig and V. Panferov, Strong solutions of the Boltzmann equation in one spatial dimension,, C. R. Acad. Sci. Paris, 342 (2006), 843. doi: 10.1016/j.crma.2006.04.005. Google Scholar [15] Kenneth M. Case, Elementary solutions of the transport equation and their applications,, Ann. Physics, 9 (1960), 1. doi: 10.1016/0003-4916(60)90060-9. Google Scholar [16] K. M. Case and P. F. Zweifel, "Linear Transport Theory,", Addison-Wesley Publishing Co., (1967). Google Scholar [17] C. Cercignani, Elementary solutions of the linearized gas-dynamics Boltzmann equation and their application to the slip-flow problem,, Ann. Physics, 20 (1962), 219. doi: 10.1016/0003-4916(62)90199-9. Google Scholar [18] C. Cercignani, Plane Couette flow according to the method of elementary solutions,, J. Math. Anal. Applic., 11 (1965), 93. doi: 10.1016/0022-247X(65)90071-5. Google Scholar [19] C. Cercignani, Methods of solution of the linearized Boltzmann equation for rarefied gas dynamics,, J. Quant. Spectrosc. Radiat. Transfer, 11 (1971), 973. doi: 10.1016/0022-4073(71)90068-9. Google Scholar [20] C. Cercignani, Analytic solution of the temperature jump problem for the BGK model,, TTSP, 6 (1977), 29. Google Scholar [21] C. Cercignani, Solution of a linearized kinetic model for an ultrarelativistic gas,, J. Stat. Phys., 42 (1986), 601. doi: 10.1007/BF01127731. Google Scholar [22] C. Cercignani, "Mathematical Methods in Kinetic Theory,", Plenum Press, (1969). Google Scholar [23] C. Cercignani, "Slow Rarefied Flows. Theory and Application to Micro-Electro-Mechanical Systems,", Progress in Mathematical Physics, 41 (2006). Google Scholar [24] C. Cercignani and F. Sernagiotto, The method of elementary solutions for time-dependent problems in linearized kinetic theory,, Ann. Physics, 30 (1964), 154. doi: 10.1016/0003-4916(64)90308-2. Google Scholar [25] Ch. Dalitz, Half-space problem of the Boltzmann equation for charged particles,, J. Stat. Phys., 88 (1997), 129. doi: 10.1007/BF02508467. Google Scholar [26] E. de Groot and Ch. Dalitz, Exact solution for a boundary value problem in semiconductor kinetic theory,, J. Math. Phys., 38 (1997), 4629. doi: 10.1063/1.532111. Google Scholar [27] L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and BGK equations,, Arch. Rat. Mech. Anal., 110 (1990), 73. doi: 10.1007/BF00375163. Google Scholar [28] L. Desvillettes and F. Salvarani, Asymptotic behavior of degenerate linear transport equations,, Bull. Sci. Math., 133 (2009), 848. doi: 10.1016/j.bulsci.2008.09.001. Google Scholar [29] A. Frangi, A. Frezzotti and S. Lorenzani, On the application of the BGK kinetic model to the analysis of gas-structure interaction in MEMS,, Computers and Structures, 85 (2007), 810. doi: 10.1016/j.compstruc.2007.01.011. Google Scholar [30] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources,, J. Comp. Phys., 229 (2010), 7625. doi: 10.1016/j.jcp.2010.06.017. Google Scholar [31] L. Gosse, Transient radiative transfer in the grey case: Well-balanced and asymptotic-preserving schemes built on Case's elementary solutions,, Journal of Quantitative Spectroscopy & Radiative Transfer, 112 (2011), 1995. doi: 10.1016/j.jqsrt.2011.04.003. Google Scholar [32] L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations,, C. R. Math. Acad. Sci. Paris, 334 (2002), 337. doi: 10.1016/S1631-073X(02)02257-4. Google Scholar [33] L. Gosse and G. Toscani, Space localization and well-balanced scheme for discrete kinetic models in diffusive regimes,, SIAM J. Numer. Anal., 41 (2003), 641. doi: 10.1137/S0036142901399392. Google Scholar [34] J. Greenberg and A. Y. Leroux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations,, SIAM J. Numer. Anal., 33 (1996), 1. doi: 10.1137/0733001. Google Scholar [35] W. Greenberg, C. V. M. van der Meea and P. F. Zweifel, Generalized kinetic equations,, Integr. Equa. Oper. Theory, 7 (1984), 60. doi: 10.1007/BF01204914. Google Scholar [36] N. Hadjiconstantinou and A. Garcia, Molecular simulations of sound wave propagation in simple gases,, Physics of Fluids, 13 (2001), 1040. doi: 10.1063/1.1352630. Google Scholar [37] E. Isaacson and B. Temple, Convergence of the $2 \times 2$ Godunov method for a general resonant nonlinear balance law,, SIAM J. Appl. Math., 55 (1995), 625. doi: 10.1137/S0036139992240711. Google Scholar [38] Shi Jin and David Levermore, The discrete-ordinate method in diffusive regimes,, Transp. Theor. Stat. Phys., 20 (1991), 413. doi: 10.1080/00411459108203913. Google Scholar [39] A. Kadir Aziz, D. A. French, S. Jensen and R. B. Kellogg, Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem,, M2AN Math. Model. Numer. Anal., 33 (1999), 895. doi: 10.1051/m2an:1999125. Google Scholar [40] H. Kaper, A constructive approach to the solution of a class of boundary value problems of mixed type,, J. Math. Anal. Applic., 63 (1978), 691. doi: 10.1016/0022-247X(78)90066-5. Google Scholar [41] H. Kaper, Boundary value problems of mixed type arising in the kinetic theory of gases,, SIAM J. Math. Anal., 10 (1979), 161. doi: 10.1137/0510017. Google Scholar [42] H. Kaper, Spectral representation of an unbounded linear transformation arising in the kinetic theory of gases,, SIAM J. Math. Anal., 10 (1979), 179. doi: 10.1137/0510018. Google Scholar [43] Tomaž Klinc, On completeness of eigenfunctions of the one-speed transport equation,, Commun. Math. Phys., 41 (1975), 273. doi: 10.1007/BF01608991. Google Scholar [44] G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases,", Interaction of Mechanics and Mathematics, (2010). Google Scholar [45] J. T. Kriese, T. S. Chang and C. E. Siewert, Elementary solutions of coupled model equations in the kinetic theory of gases,, Int. J. Eng. Sci., 12 (1974), 441. doi: 10.1016/0020-7225(74)90064-0. Google Scholar [46] A. V. Latyshev, The use of Case's method to solve the linearized BGK equations for the temperature-jump problem,, J. Appl. Math. Mech., 54 (1990), 480. doi: 10.1016/0021-8928(90)90059-J. Google Scholar [47] A. V. Latyshev and A. A. Yushkanov, An analytic solution of the problem of temperature and density jumps of a vapor over a surface in the presence of a temperature gradient,, J. Applied Math. Mech., 58 (1994), 259. doi: 10.1016/0021-8928(94)90054-X. Google Scholar [48] Ph. LeFloch and A. E. Tzavaras, Representation of weak limits and definition of nonconservative products,, SIAM J. Math. Anal., 30 (1999), 1309. doi: 10.1137/S0036141098341794. Google Scholar [49] T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation,, Physica D, 188 (2004), 178. doi: 10.1016/j.physd.2003.07.011. 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##### References:
 [1] D. Amadori, L. Gosse and G. Guerra, Global BV entropy solutions and uniqueness for hyperbolic systems of balance laws,, Arch. Rational Mech. Anal., 162 (2002), 327. doi: 10.1007/s002050200198. Google Scholar [2] K. Aoki and C. Cercignani, A technique for time-dependent boundary value problems in the kinetic theory of gases. I. Basic analysis,, Z. Angew. Math. Phys., 35 (1984), 127. doi: 10.1007/BF00947927. Google Scholar [3] J. Appell, A. S. Kalitvin and P. P. Zabrejko, Boundary value problems for integro-differential equations of Barbashin type,, J. Integral Equ. Applic., 6 (1994), 1. doi: 10.1216/jiea/1181075787. Google Scholar [4] A. Arnold, J. A. Carrillo and M. D. Tidriri, Large-time behavior of discrete equations with non-symmetric interactions,, Math. Mod. Meth. in Appl. Sci., 12 (2002), 1555. Google Scholar [5] C. Bardos, F. Golse and D. Levermore, Fluid dynamic limits of kinetic equations. I. Formal derivations,, J. Stat. Phys., 63 (1991), 323. doi: 10.1007/BF01026608. Google Scholar [6] L. B. Barichello, M. Camargo, P. Rodrigues and C. E. Siewert, Unified solutions to classical flow problems based on the BGK model,, Z. Angew. Math. Phys., 52 (2001), 517. doi: 10.1007/PL00001559. Google Scholar [7] L. B. Barichello and C. E. Siewert, A discrete-ordinates solution for a non-grey model with complete frequency redistribution,, JQSRT, 62 (1999), 665. Google Scholar [8] G. R. Bart and R. L. Warnock, Linear integral equations of the third kind,, SIAM J. Math. Anal., 4 (1973), 609. doi: 10.1137/0504053. Google Scholar [9] P. Bassanini, C. Cercignani and C. D. Pagani, Comparison of kinetic theory analyses of linearized heat transfer between parallel plates,, Int. J. Heat Mass Transfer, 10 (1967), 447. doi: 10.1016/0017-9310(67)90165-2. Google Scholar [10] P. Bassanini, C. Cercignani and C. D. Pagani, Influence of the accommodation coefficient on the heat transfer in a rarefied gas,, Int. J. Heat Mass Transfer, 11 (1968), 1359. doi: 10.1016/0017-9310(68)90181-6. Google Scholar [11] R. Beals, An abstract treatment of some forward-backward problems of transport and scattering,, J. Funct. Anal., 34 (1979), 1. doi: 10.1016/0022-1236(79)90021-1. Google Scholar [12] R. Beals and V. Protopopescu, Half-range completeness for the Fokker-Planck equation,, J. Stat. Phys., 32 (1983), 565. doi: 10.1007/BF01008957. Google Scholar [13] M. Bennoune, M. Lemou and L. Mieussens, Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier-Stokes asymptotics,, J. Comp. Phys., 227 (2008), 3781. doi: 10.1016/j.jcp.2007.11.032. Google Scholar [14] A. Biryuk, W. Craig and V. Panferov, Strong solutions of the Boltzmann equation in one spatial dimension,, C. R. Acad. Sci. Paris, 342 (2006), 843. doi: 10.1016/j.crma.2006.04.005. Google Scholar [15] Kenneth M. Case, Elementary solutions of the transport equation and their applications,, Ann. Physics, 9 (1960), 1. doi: 10.1016/0003-4916(60)90060-9. Google Scholar [16] K. M. Case and P. F. Zweifel, "Linear Transport Theory,", Addison-Wesley Publishing Co., (1967). Google Scholar [17] C. Cercignani, Elementary solutions of the linearized gas-dynamics Boltzmann equation and their application to the slip-flow problem,, Ann. Physics, 20 (1962), 219. doi: 10.1016/0003-4916(62)90199-9. Google Scholar [18] C. Cercignani, Plane Couette flow according to the method of elementary solutions,, J. Math. Anal. Applic., 11 (1965), 93. doi: 10.1016/0022-247X(65)90071-5. Google Scholar [19] C. Cercignani, Methods of solution of the linearized Boltzmann equation for rarefied gas dynamics,, J. Quant. Spectrosc. Radiat. Transfer, 11 (1971), 973. doi: 10.1016/0022-4073(71)90068-9. Google Scholar [20] C. Cercignani, Analytic solution of the temperature jump problem for the BGK model,, TTSP, 6 (1977), 29. Google Scholar [21] C. Cercignani, Solution of a linearized kinetic model for an ultrarelativistic gas,, J. Stat. Phys., 42 (1986), 601. doi: 10.1007/BF01127731. Google Scholar [22] C. Cercignani, "Mathematical Methods in Kinetic Theory,", Plenum Press, (1969). Google Scholar [23] C. Cercignani, "Slow Rarefied Flows. Theory and Application to Micro-Electro-Mechanical Systems,", Progress in Mathematical Physics, 41 (2006). Google Scholar [24] C. Cercignani and F. Sernagiotto, The method of elementary solutions for time-dependent problems in linearized kinetic theory,, Ann. Physics, 30 (1964), 154. doi: 10.1016/0003-4916(64)90308-2. Google Scholar [25] Ch. Dalitz, Half-space problem of the Boltzmann equation for charged particles,, J. Stat. Phys., 88 (1997), 129. doi: 10.1007/BF02508467. Google Scholar [26] E. de Groot and Ch. Dalitz, Exact solution for a boundary value problem in semiconductor kinetic theory,, J. Math. Phys., 38 (1997), 4629. doi: 10.1063/1.532111. Google Scholar [27] L. Desvillettes, Convergence to equilibrium in large time for Boltzmann and BGK equations,, Arch. Rat. Mech. Anal., 110 (1990), 73. doi: 10.1007/BF00375163. Google Scholar [28] L. Desvillettes and F. Salvarani, Asymptotic behavior of degenerate linear transport equations,, Bull. Sci. Math., 133 (2009), 848. doi: 10.1016/j.bulsci.2008.09.001. Google Scholar [29] A. Frangi, A. Frezzotti and S. Lorenzani, On the application of the BGK kinetic model to the analysis of gas-structure interaction in MEMS,, Computers and Structures, 85 (2007), 810. doi: 10.1016/j.compstruc.2007.01.011. Google Scholar [30] F. Filbet and S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources,, J. Comp. Phys., 229 (2010), 7625. doi: 10.1016/j.jcp.2010.06.017. Google Scholar [31] L. Gosse, Transient radiative transfer in the grey case: Well-balanced and asymptotic-preserving schemes built on Case's elementary solutions,, Journal of Quantitative Spectroscopy & Radiative Transfer, 112 (2011), 1995. doi: 10.1016/j.jqsrt.2011.04.003. Google Scholar [32] L. Gosse and G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations,, C. R. Math. Acad. Sci. Paris, 334 (2002), 337. doi: 10.1016/S1631-073X(02)02257-4. Google Scholar [33] L. Gosse and G. Toscani, Space localization and well-balanced scheme for discrete kinetic models in diffusive regimes,, SIAM J. Numer. Anal., 41 (2003), 641. doi: 10.1137/S0036142901399392. Google Scholar [34] J. Greenberg and A. Y. Leroux, A well balanced scheme for the numerical processing of source terms in hyperbolic equations,, SIAM J. Numer. Anal., 33 (1996), 1. doi: 10.1137/0733001. Google Scholar [35] W. Greenberg, C. V. M. van der Meea and P. F. Zweifel, Generalized kinetic equations,, Integr. Equa. Oper. Theory, 7 (1984), 60. doi: 10.1007/BF01204914. Google Scholar [36] N. Hadjiconstantinou and A. Garcia, Molecular simulations of sound wave propagation in simple gases,, Physics of Fluids, 13 (2001), 1040. doi: 10.1063/1.1352630. Google Scholar [37] E. Isaacson and B. Temple, Convergence of the $2 \times 2$ Godunov method for a general resonant nonlinear balance law,, SIAM J. Appl. Math., 55 (1995), 625. doi: 10.1137/S0036139992240711. Google Scholar [38] Shi Jin and David Levermore, The discrete-ordinate method in diffusive regimes,, Transp. Theor. Stat. Phys., 20 (1991), 413. doi: 10.1080/00411459108203913. Google Scholar [39] A. Kadir Aziz, D. A. French, S. Jensen and R. B. Kellogg, Origins, analysis, numerical analysis, and numerical approximation of a forward-backward parabolic problem,, M2AN Math. Model. Numer. Anal., 33 (1999), 895. doi: 10.1051/m2an:1999125. Google Scholar [40] H. Kaper, A constructive approach to the solution of a class of boundary value problems of mixed type,, J. Math. Anal. Applic., 63 (1978), 691. doi: 10.1016/0022-247X(78)90066-5. Google Scholar [41] H. Kaper, Boundary value problems of mixed type arising in the kinetic theory of gases,, SIAM J. Math. Anal., 10 (1979), 161. doi: 10.1137/0510017. Google Scholar [42] H. Kaper, Spectral representation of an unbounded linear transformation arising in the kinetic theory of gases,, SIAM J. Math. Anal., 10 (1979), 179. doi: 10.1137/0510018. Google Scholar [43] Tomaž Klinc, On completeness of eigenfunctions of the one-speed transport equation,, Commun. Math. Phys., 41 (1975), 273. doi: 10.1007/BF01608991. Google Scholar [44] G. M. Kremer, "An Introduction to the Boltzmann Equation and Transport Processes in Gases,", Interaction of Mechanics and Mathematics, (2010). Google Scholar [45] J. T. Kriese, T. S. Chang and C. E. Siewert, Elementary solutions of coupled model equations in the kinetic theory of gases,, Int. J. Eng. Sci., 12 (1974), 441. doi: 10.1016/0020-7225(74)90064-0. Google Scholar [46] A. V. Latyshev, The use of Case's method to solve the linearized BGK equations for the temperature-jump problem,, J. Appl. Math. Mech., 54 (1990), 480. doi: 10.1016/0021-8928(90)90059-J. Google Scholar [47] A. V. Latyshev and A. A. Yushkanov, An analytic solution of the problem of temperature and density jumps of a vapor over a surface in the presence of a temperature gradient,, J. Applied Math. Mech., 58 (1994), 259. doi: 10.1016/0021-8928(94)90054-X. Google Scholar [48] Ph. LeFloch and A. E. Tzavaras, Representation of weak limits and definition of nonconservative products,, SIAM J. Math. Anal., 30 (1999), 1309. doi: 10.1137/S0036141098341794. Google Scholar [49] T.-P. Liu, T. Yang and S.-H. Yu, Energy method for Boltzmann equation,, Physica D, 188 (2004), 178. doi: 10.1016/j.physd.2003.07.011. 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